| L(s) = 1 | + (1.59 + 1.83i)2-s + (−1.87 − 1.20i)3-s + (−0.272 + 1.89i)4-s + (−2.69 − 1.22i)5-s + (−0.771 − 5.36i)6-s + (−1.33 + 4.53i)7-s + (4.26 − 2.73i)8-s + (−1.67 − 3.67i)9-s + (−2.02 − 6.90i)10-s + (11.0 + 9.55i)11-s + (2.79 − 3.22i)12-s + (−22.7 + 6.68i)13-s + (−10.4 + 4.77i)14-s + (3.56 + 5.54i)15-s + (19.1 + 5.63i)16-s + (12.1 − 1.75i)17-s + ⋯ |
| L(s) = 1 | + (0.796 + 0.919i)2-s + (−0.624 − 0.401i)3-s + (−0.0682 + 0.474i)4-s + (−0.538 − 0.245i)5-s + (−0.128 − 0.894i)6-s + (−0.190 + 0.648i)7-s + (0.532 − 0.342i)8-s + (−0.186 − 0.408i)9-s + (−0.202 − 0.690i)10-s + (1.00 + 0.868i)11-s + (0.233 − 0.269i)12-s + (−1.75 + 0.513i)13-s + (−0.747 + 0.341i)14-s + (0.237 + 0.369i)15-s + (1.19 + 0.352i)16-s + (0.716 − 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.991784 + 0.337266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991784 + 0.337266i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-5.15 + 22.4i)T \) |
| good | 2 | \( 1 + (-1.59 - 1.83i)T + (-0.569 + 3.95i)T^{2} \) |
| 3 | \( 1 + (1.87 + 1.20i)T + (3.73 + 8.18i)T^{2} \) |
| 5 | \( 1 + (2.69 + 1.22i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (1.33 - 4.53i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-11.0 - 9.55i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (22.7 - 6.68i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (-12.1 + 1.75i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (-5.54 - 0.797i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (3.08 + 21.4i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (0.0511 - 0.0328i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (-13.3 + 6.09i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-4.95 + 10.8i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (27.4 - 42.6i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 + 23.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (28.2 - 96.2i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-34.0 + 9.99i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (49.9 + 77.7i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (72.5 - 62.8i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (-35.6 - 41.1i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-6.81 + 47.3i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (10.2 + 34.9i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-95.0 + 43.4i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (2.45 - 3.81i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-119. - 54.6i)T + (6.16e3 + 7.11e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.36412889734448636649126248200, −16.49992821706803998327430328622, −15.09933363105423117212647243843, −14.40282234487935757105808770181, −12.45317127327953400175966910430, −11.96481455974810033292033752014, −9.595543649412257594714026474741, −7.40468695588340325545011944009, −6.21292927699439652072709677990, −4.62795197390074098227094132564,
3.51707798217482505629092660775, 5.17654399874535466173063186251, 7.57682152792827381472437539532, 10.08083662358893388906891451374, 11.26840126642299909379336250808, 12.05037109554911537186375094591, 13.56045456150962152083653967352, 14.72252048251105479410591442193, 16.53755189726705494040137531512, 17.18822784271308461776725320586
